2018-10-01_Physics_For_You

1

Z

is known as admittance (Y). erefore,

Y

Z

RjL

RL

== jC

−

+

+

1

222

ω

ω

ω =

++−

+

RjCR LC L

RL

()ωωω

ω

232

222

• Resonance Condition
  e magnitude of the admittance,

YY

RCRLCL

RL

==

++−

+

||

(^22) () 32 2
222
ωωω
ω
e admittance will be minimum, when
ZCR^2 + Z^3 L^2 C – ZL = 0
ω= −

1 2

LC^2

R

L

It gives the condition of resonance and the

corresponding frequency, ν

ω

ππ

== −

2

1

2

1 2

LC^2

R

L

is known as resonance frequency. At resonance

frequency admittance is minimum or the

impedance is maximum. us, the parallel circuit

does not allow this frequency from the source to

pass in the circuit. Due to this reason the circuit

with such a frequency is known as rejector circuit.

If R = 0, resonance frequency is

1

2 π LC

same as

resonance frequency in series circuit.

At resonance, the reactive component of Y is real.

e reciprocal of the admittance is called the parallel

resistor or the dynamic resistance. e dynamic

resistance is thus, reciprocal of the real part of the

admittance.

Dynamic resistance=

RL+

R

(^22) ω 2
Substitutingω^2
2
2

1

=−

LC

R

L

we have, dynamic resistance= L

CR

? peak current through the supply

==

V

LCR

VCR

L

00

/

e peak current through capacitor

==

V

C

(^0) VC 0
1/

.

ω

ω

e ratio of the peak current through capacitor and

through the supply is known as Q-factor.

us, Q-factor==

VC

VCRL

L

R

0

0

ω ω

/

is is basically the measure of current

magnication. e rejector circuit at resonance

exhibits current magnication of ωL

R

,similar to the

voltage magnication of the same ratio exhibited by

the series acceptor circuit at resonance.

At resonance the current through the supply and

voltage are in phase, while the current through the

capacitor leads the voltage by 90°.

LC OSCILLATIONS

• L

dq

dt

q

C

I

dq

dt

2

2 +=^0 =−







' 

• Frequency,
  υ
  π

=

1

2 LC

• Oscillation of charge, q = qmcos(Zt + φ)

(where Z = natural frequency =

1

LC

)

• Oscillation of current, I = Imsin(Zt + φ)
  (where Im = Zqm)


• Electric energy, U

q

C

q

C

E==m t+

2 2

2

22

cos(ωφ)

• Magnetic energy,

ULILqt

q

C

Bm== +=m t+

1

2

1

22

2222

2

ωωsin( φω)sin^2 ()φ

TRANSFORMERS

• A transformer consists of an iron core on which
  a primary coil of turns Np and a secondary coil of
  turns NS are bound. If the primary coil is connected
  to an AC source, the primary and secondary
  voltages and current are related by

V

N

N

s s V

p

= p









; I

N

N

s p I

s

= p









• If the secondary coil has a greater number of
  turns than the primary, the voltage is stepped-up
  (Vs > Vp). is type of arrangement is called a
  step-up transformer. If the secondary coil has
  turns less than the primary, we have a step-down
  transformer.


• Eciency of transformer,

η= =

Outputpower

Inputpower

VI

VI

ss

pp

For an ideal transformer, K = 1

• e ratio NP/NS is known as turns ratio of the
  transformer. Transformers are mainly characterized
  by the turns ratio.